Monday, April 29, 2013

Before we begin, I'd like to thank listener Bret McDermitt for posting another nice review on iTunes. I'd also like to thank the Math Insider website for featuring Math Mutation in an article on great math podcasts-- check the article out at the link in the show notes! Now, let's get on to today's topic.

As I looked out the window recently on a windy day, watching leaves and dirt fly around outside, it brought to mind Rudy Rucker's concept of a 'paracomputer', a computer based on observing natural processes that happen to be equivalent to a natural computation. Let's review the background of this idea briefly so we can explain it.

You may recall from some earlier podcasts the ideas of Stephen Wolfram's controversial tome "A New Kind of Science", which claims that we can view many natural phenomena as similar to a computation such as a cellular automaton. One popular example of a cellular automaton is Conway's Game of Life. In this game, each square in a grid can be "alive" or "dead", usually represented by being colored black or white. A live square stays alive for the next time step if and only if it has exactly two or three live neighbors, otherwise dying of loneliness or overcrowding. And a dead square is "born" and becomes alive if it has exactly three live neighbors. The amazing thing about this simple game is that depending on the initial set of live squares, it can display a huge range of lifelike behaviors, as you can see illustrated on the Wikipedia page linked in the show notes. This led researchers such as Wolfram to hypothesize that such simple computations might explain the basics of how our universe really works.

If you look at the patterns created by various Life configurations, you can see three basic types. Predictable patterns settle into an endless repetition of simple elements, such as a square group of cells simply living forever in the middle. Chaotic patterns end up looking totally random, like the static on a malfunctioning TV set. The most interesting ones are what Rudy Rucker calls "gnarly" patterns, which seem to have interesting structure but are unpredictable. These gnarly patterns are what seem to have promise to represent real life: for example, a gnarly pattern might look analogous to clouds moving in the sky, the life cycle of a jellyfish, or celestial bodies interacting over the lifetime of a solar system. These all share the property that their ultimate results are in theory predictable, but so complex to calculate that there's no fundamentally better way to figure out what will happen than to wait and see.

After observing many of these gnarly systems, Wolfram hypothesized the Principle of Computational Equivalence. This states that at some fundamental level, all these gnarly real-life systems are each capable of acting as a universal computer, and thus equivalent to each other. So, for example, by coming up with a suitable "mapping" of clouds that represents the initial state of a jellyfish being born, and somehow manipulating the clouds into this configuration, you would be able to, by watching those clouds, predict exactly what is happening at any stage of the jellyfish's life. You just need to decode the current state of the clouds into the equivalent state of the jellyfish.

And this is Rudy Rucker's idea of a "paracomputer". The concept is that you figure out some gnarly physical phenomenon that you can control, identify the proper mapping to and from whatever other type of system you want to analyze, and just set it in motion. Sounds pretty cool, right? So, all those old fortune-tellers who claim to be able to tell the future from tea leaves might actually have the core of a feasible idea: all you have to do is figure out enough detail about your life to map its precise state into the tea leaves in a very large kettle, shake the kettle in a defined way so the leaves will represent each future state of your life, read the results in the leaves, and map the final leaf pattern to the details of your life. Presto! You now know every detail of your future.

Now if you're picky, you might have spotted a few flaws in this idea. Wolfram's Principle itself is very controversial, as it seems to collapse what could potentially be many different complexity classes, and claim that a huge range of phenomena are equally complex to compute. His observations provide some evidence, but nothing close to a real proof. There's also the question of how you would fully figure out the starting state of your life, to map to the tea leaves: you might object as the fortune-teller takes out a bone saw, opens up your skull, and carefully extracts the state of each neuron in your brain. Even after your brain has been carefully put back together without modifying the state, then there's the question of how many tea leaves it would take to map your life: the fortune-teller might need all the tea in China to properly represent it, especially if you live the exciting life of a math podcast fan. And finally, there's the question of how long it would take the tea leaves to reach their representation of any point in your future: would this really be any faster than just waiting for your future to happen?

But, nevertheless, it is kind of fun to think that nature really consists of computers all around us, just waiting to be properly understood. Maybe soon you'll be able to turn in your PlayStation 3 and play Call Of Duty with leaves blowing in the wind in your backyard.

Note that this podcast is intended mainly for audio consumption, so you will not see the numerous illustrations & diagrams you would find at most math sites, though these are linked in the show notes whenever possible.

The Math Mutation Book

The Math Mutation book, "Math Mutation Classics: Exploring Interesting, Fun, and Weird Corners of Mathematics", is now available. You can order it from Amazon at this link.

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